Subgroups — C₃×#10601698 #14287413

Order 1920.

304 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#65313	C₂	1	2	yes	maximal
3	C₃×Q₈⋊#168	C₂	1	2	yes	maximal
4	C₃×#64678	C₂	1	1	yes	maximal, characteristic
5	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic
6	#10601698	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₄	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₂²	1	2	yes	—
9	C₃×C₅×#107	C₄	1	1	yes	characteristic
10	C₃×C₅×#103	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊#168	—	2	2	no	—
13	C₃×C₄⋊#168	C₂²	1	2	yes	—
14	C₃×#3935	—	2	2	no	—
15	C₃×C₄×#168	C₂²	1	1	yes	characteristic
16	C₃×C₄×#168	C₂²	1	1	yes	characteristic
17	C₃×#1591	—	5	1	no	maximal, Hall
18	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
19	#65313	C₂×C₃	1	2	yes	—
20	Q₈⋊#168	C₂×C₃	1	2	yes	—
21	#64678	C₂×C₃	1	1	yes	characteristic
22	C₈×#168	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
24	C₃×C₅×Q₁₆	—	2	2	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	—	2	2	no	—
28	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×#168	—	2	1	no	—
32	C₂×C₃×#168	—	2	2	no	—
33	C₂×C₃×#168	D₄	1	1	yes	characteristic
34	C₂×C₃×#168	D₄	1	1	yes	characteristic
35	C₃×#446	—	5	2	no	—
36	C₃×C₄×Q₁₆	D₅	1	1	yes	characteristic
37	C₃×#335	—	5	1	no	—
38	C₃×Q₈⋊C₈	—	5	2	no	—
39	C₃×C₈²	—	5	1	no	—
40	C₂×C₅×Q₁₆	C₃×C₄	1	1	yes	characteristic
41	C₄×C₅×Q₈	C₂²×C₃	1	2	yes	—
42	C₅×#107	C₃×C₄	1	1	yes	characteristic
43	C₅×#103	C₃×C₄	1	2	yes	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₄⋊#168	—	2	2	no	—
46	C₄⋊#168	C₂²×C₃	1	2	yes	—
47	#3935	—	2	2	no	—
48	C₄×#168	C₂²×C₃	1	1	yes	characteristic
49	C₄×#168	C₂²×C₃	1	1	yes	characteristic
50	#1591	—	5	1	no	Sylow
51	C₃×C₅×Q₈	—	2	2	no	—
52	C₃×C₅×Q₈	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
58	C₃×C₅×C₈	—	2	1	no	—
59	C₃×C₅×C₈	M₁₆	1	2	yes	—
60	C₃×#168	—	2	2	no	—
61	C₃×#168	—	2	1	no	—
62	C₃×#168	—	4	2	no	—
63	C₃×#168	Q₁₆	1	2	yes	complemented
64	C₂×C₃×Q₁₆	Dic₅	1	1	yes	characteristic
65	C₃×C₄×Q₈	C₂×D₅	1	2	yes	—
66	C₃×#107	Dic₅	1	1	yes	characteristic
67	C₃×C₄⋊C₈	—	5	2	no	—
68	C₃×C₄⋊C₈	—	10	2	no	—
69	C₃×#103	Dic₅	1	2	yes	—
70	C₃×#97	—	10	2	no	—
71	C₃×C₄×C₈	—	5	1	no	—
72	C₃×C₄×C₈	—	5	1	no	—
73	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
75	C₅×Q₁₆	—	2	2	no	—
76	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	2	2	no	—
79	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
80	C₄²×C₅	—	2	2	no	—
81	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
82	C₂×#168	—	2	1	no	—
83	C₂×#168	—	2	2	no	—
84	C₂×#168	C₃×D₄	1	1	yes	characteristic
85	C₂×#168	C₃×D₄	1	1	yes	characteristic
86	#446	—	5	2	no	—
87	C₄×Q₁₆	C₃×D₅	1	1	yes	characteristic
88	#335	—	5	1	no	—
89	Q₈⋊C₈	—	5	2	no	—
90	C₈²	—	5	1	no	—
91	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	4	2	no	—
94	C₃×C₄×C₅	—	4	2	no	—
95	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
96	C₃×C₄×C₅	#131	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
98	C₃×C₄×C₅	#135	1	1	yes	characteristic
99	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
100	C₃×Q₁₆	—	2	2	no	—
101	C₂×C₃×C₈	—	5	1	no	—
102	C₂×C₃×C₈	—	5	1	no	—
103	C₂×C₃×C₈	—	10	1	no	—
104	C₂×C₃×C₈	—	10	2	no	—
105	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
111	C₅×Q₈	—	2	2	no	—
112	C₅×Q₈	—	2	2	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
118	C₅×C₈	—	2	1	no	—
119	C₅×C₈	C₃×M₁₆	1	2	yes	—
120	#168	—	2	2	no	—
121	#168	—	2	1	no	—
122	#168	—	4	2	no	—
123	#168	C₃×Q₁₆	1	2	yes	complemented
124	C₂×Q₁₆	C₃×Dic₅	1	1	yes	characteristic
125	C₄×Q₈	C₂×C₃×D₅	1	2	yes	—
126	#107	C₃×Dic₅	1	1	yes	characteristic
127	C₄⋊C₈	—	5	2	no	—
128	C₄⋊C₈	—	10	2	no	—
129	#103	C₃×Dic₅	1	2	yes	—
130	#97	—	10	2	no	—
131	C₄×C₈	—	5	1	no	—
132	C₄×C₈	—	5	1	no	—
133	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
134	C₂×C₃×C₅	#436	1	1	yes	characteristic
135	C₂×C₃×C₅	C₄×Q₁₆	1	1	yes	characteristic
136	C₂×C₃×C₅	#443	1	1	yes	characteristic
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
142	C₂×C₃×C₄	#711	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	5	2	no	—
146	C₃×C₈	—	10	2	no	—
147	C₃×C₈	—	10	1	no	—
148	C₃×C₈	—	20	2	no	—
149	C₃×C₈	#681	1	2	yes	—
150	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	4	2	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
155	C₄×C₅	C₃×#131	1	1	yes	characteristic
156	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
157	C₄×C₅	C₃×#135	1	1	yes	characteristic
158	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
159	Q₁₆	—	2	2	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	10	1	no	—
163	C₂×C₈	—	10	2	no	—
164	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
165	C₂×C₈	C₂×C₃×Dic₅	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	C₂×C₃×Dic₅	1	2	yes	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
170	C₃×C₅	#1591	1	1	yes	characteristic, complemented, Hall
171	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	4	2	no	—
174	C₃×C₄	—	4	2	no	—
175	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
176	C₃×C₄	#4065	1	1	yes	characteristic
177	C₃×C₄	C₂×#681	1	1	yes	characteristic
178	C₃×C₄	#4094	1	1	yes	characteristic
179	C₂×C₅	C₃×#436	1	1	yes	characteristic
180	C₂×C₅	C₃×C₄×Q₁₆	1	1	yes	characteristic
181	C₂×C₅	C₃×#443	1	1	yes	characteristic
182	Q₈	—	2	2	no	—
183	Q₈	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
187	C₂×C₄	C₃×#711	1	1	yes	characteristic
188	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	5	2	no	—
191	C₈	—	10	2	no	—
192	C₈	—	10	1	no	—
193	C₈	—	20	2	no	—
194	C₈	C₃×#681	1	2	yes	—
195	C₂×C₃	#65301	1	1	yes	characteristic
196	C₂×C₃	Q₁₆×Dic₅	1	1	yes	characteristic
197	C₂×C₃	#65485	1	1	yes	characteristic
198	C₅	C₃×#1591	1	1	yes	characteristic, complemented, Sylow
199	C₂²	C₃×D₄×Dic₅	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	C₃×D₅×Q₁₆	1	1	yes	characteristic
204	C₄	C₃×#4065	1	1	yes	characteristic
205	C₄	C₂×C₃×#681	1	1	yes	characteristic
206	C₄	C₃×#4094	1	1	yes	characteristic
207	C₃	#10601698	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	C₃×#65301	1	1	yes	characteristic
209	C₂	C₃×Q₁₆×Dic₅	1	1	yes	characteristic
210	C₂	C₃×#65485	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×Q₁₆	C₂
socle, prime radical	C₂²×C₃×C₅	C₄×D₄
cosocle	C₄²×C₅	C₂³×C₃
hypercenter	C₃×C₄×Q₁₆	D₅
hypocenter	C₅	C₃×#1591
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×C₄×Q₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
6. #10601698	C₃	direct (N × H)
63. C₃×#168	Q₁₆	semidirect
123. #168	C₃×Q₁₆	semidirect
170. C₃×C₅	#1591	semidirect
198. C₅	C₃×#1591	semidirect
207. C₃	#10601698	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₄×C₅×Q₁₆	2
C₃×#65313	2
C₃×Q₈⋊#168	2
C₃×#64678	2
C₃×C₈×#168	2
#10601698	3
C₃×#1591	5

Subgroups — C3×#10601698 #14287413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10601698 #14287413